/*
 * Copyright (c) 1994, 2013, Oracle and/or its affiliates. All rights reserved.
 * ORACLE PROPRIETARY/CONFIDENTIAL. Use is subject to license terms.
 *
 *
 *
 *
 *
 *
 *
 *
 *
 *
 *
 *
 *
 *
 *
 *
 *
 *
 *
 *
 */

package java.lang;

import java.lang.annotation.Native;

/**
 * <p>Integer类包装了一个原始类型int的值在一个对象中。
 * 一个Integer类的对象包含了一个单一的int类型的字段。
 * 
 * <p>此外，这个类提供了几个将int转为String和将String转为int的方法，
 * 还有一些在处理int类型时有用的常量和方法。
 * 
 * <p>实现的注解：位处理的实现方法（例如highestOneBit，numberOfTrailingZeros，numberOfTrailingZeros）
 * 是基于Henry S. Warren, Jr.'s Hacker's Delight的材料。
 *
 * @author  Lee Boynton
 * @author  Arthur van Hoff
 * @author  Josh Bloch
 * @author  Joseph D. Darcy
 * @since JDK1.0
 */
public final class Integer extends Number implements Comparable<Integer> {
    /**
     * 一个持有int类型的最小值的常量，-2<sup>31</sup>，-2147483648
     */
    @Native public static final int   MIN_VALUE = 0x80000000;
    //1后面3+7*4=31个0，对应2的31次方（1后面3个0对应1000对应8对应2的3次方）

    /**
     * 一个持有int类型的最大值的常量，2<sup>31</sup>-1，2147483647 
     */
    @Native public static final int   MAX_VALUE = 0x7fffffff;
    //3+7*4=31个1，对应2的31次方-1（3个1对应111对应7对应2的3次方-1）

    /**
     * 代表着原始类型int的Class实例，类似int.class 原始类
     *
     * @since   JDK1.1
     */
    @SuppressWarnings("unchecked")
    public static final Class<Integer>  TYPE = (Class<Integer>) Class.getPrimitiveClass("int");

    /**
     * 所有可能作为一个String代表数字的char，0-9和a-z，因为可能出现2-35进制。
     */
    final static char[] digits = {
        '0' , '1' , '2' , '3' , '4' , '5' ,
        '6' , '7' , '8' , '9' , 'a' , 'b' ,
        'c' , 'd' , 'e' , 'f' , 'g' , 'h' ,
        'i' , 'j' , 'k' , 'l' , 'm' , 'n' ,
        'o' , 'p' , 'q' , 'r' , 's' , 't' ,
        'u' , 'v' , 'w' , 'x' , 'y' , 'z'
    };

    /**
     * <p>返回一个代表第一个参数的字符串，进制由第二个参数指定。
     * 
     * <p>如果进制小于Character.MIN_RADIX或者大于Character.MAX_RADIX，
     * 就使用10进制。
     * 
     * <p>如果第一个参数是负的，结果的第一个元素是ASCII的负号{@code '-'}({@code '\u005Cu002D'})。
     * 如果第一个参数是非负的，结果不会显示负号字符。
     * 
     * <p>结果剩余的字符代表了第一个参数的大小。
     * 如果大小是0，它由一个单独的0字符{@code '0'}({@code '\u005Cu0030'})代表。
     * 否则，大小的代表的首字符不会是字符0.
     * 会使用下面的ASCII字符作为数字：
     * 
     * <blockquote>
     *   {@code 0123456789abcdefghijklmnopqrstuvwxyz}
     * </blockquote>
     * 
     * <p>它们是从{@code '\u005Cu0030'}到{@code '\u005Cu0039'}和
     * {@code '\u005Cu0061'}到{@code '\u005Cu007A'}.
     * 如果进制是N，那么这些字符的前N个被按显示的顺序，对应与数字N。
     * 因此，对于16进制的数字是{@code 0123456789abcdef}.（这16个字符，第几个就对应数字几，从0开始，第15个f对应15）
     * 如果要求大写的字母，可以对结果调用String.toUpperCase()
     *
     * <blockquote>
     *  {@code Integer.toString(n, 16).toUpperCase()}
     * </blockquote>
     *
     * @param   i       an integer to be converted to a string.
     * @param   radix   the radix to use in the string representation.
     * @return  a string representation of the argument in the specified radix.
     * @see     java.lang.Character#MAX_RADIX
     * @see     java.lang.Character#MIN_RADIX
     */
    public static String toString(int i, int radix) {
        if (radix < Character.MIN_RADIX || radix > Character.MAX_RADIX)
        	//如果进制超过范围，则变为10进制
            radix = 10;

        /* 如果是10进制，使用更快的版本，integer本身的toString方法*/
        if (radix == 10) {
            return toString(i);
        }

        char buf[] = new char[33];  //存放结果的char数组
        boolean negative = (i < 0);
        int charPos = 32;  //写入char数组的指针，从最后一格写入

        if (!negative) {
        	//把i变成负数
            i = -i;
        }

        while (i <= -radix) {
        	//当i<=radix，例如i<=16,此时i还能被radix除
            buf[charPos--] = digits[-(i % radix)];  //charPos的位置的结果是digits数组的下标为-(i mod radix)
            i = i / radix;  
        }
        buf[charPos] = digits[-i];  //此时-radix<i<0

        if (negative) {
            buf[--charPos] = '-';  //设置负号
        }
        //创建新的string，开始位置为charPos，长度为33 - charPos
        return new String(buf, charPos, (33 - charPos));
    }

    /**
     * <p>返回一个代表无符号整数的第一个参数的字符串，进制由第二个参数指定。
     * 
     * <p>如果进制小于Character.MIN_RADIX或者大于Character.MAX_RADIX，
     * 就使用10进制。
     * 
     * <p>注意：因为第一个参数被视为无符号整数，不会打印任何符号（正负号）
     * 
     * <p>如果大小是0，它由一个单独的0字符{@code '0'}({@code '\u005Cu0030'})代表。
     * 否则，大小的代表的首字符不会是字符0.
     * 
     * <p>radix的行为和用作数字的字符与toString(int, int)一样。
     *
     * @param   i       an integer to be converted to an unsigned string.
     * @param   radix   the radix to use in the string representation.
     * @return  an unsigned string representation of the argument in the specified radix.
     * @see     #toString(int, int)
     * @since 1.8
     */
    public static String toUnsignedString(int i, int radix) {
    	//先将i变成无符号的long，再转为string
        return Long.toUnsignedString(toUnsignedLong(i), radix);
    }

    /**
     * <p>返回一个代表整数参数的字符串，字符串代表的整数是16进制的无符号integer
     * 
     * <p>如果参数是负数，返回的无符号整数是参数+2<sup>32</sup>。
     * 否则，它与参数相同。值被转换为开头没有额外的0的，16进制的ASCII数字构成的字符串。
     * 
     * <p>参数的值能被恢复，通过调用 Integer.parseUnsignedInt(s, 16)
     * 
     * <p>如果无符号整数大小为0，它由一个单独的0字符表示。{@code '0'} ({@code '\u005Cu0030'})
     * 否则，返回的无符号整数的第一个字符不会是0字符。
     * 下面的字符用作16进制的数字：
     * 
     * <blockquote>
     *  {@code 0123456789abcdef}
     * </blockquote>
     * 
     * <p>它们是从{@code '\u005Cu0030'}到{@code '\u005Cu0039'}和
     * {@code '\u005Cu0061'}到{@code '\u005Cu007A'}.
     * 如果要求大写的字母，可以调用Integer.toHexString(n).toUpperCase()
     *
     *
     * @param   i   an integer to be converted to a string.
     * @return  the string representation of the unsigned integer value
     *          represented by the argument in hexadecimal (base&nbsp;16).
     * @see #parseUnsignedInt(String, int)
     * @see #toUnsignedString(int, int)
     * @since   JDK1.0.2
     */
    public static String toHexString(int i) {
        return toUnsignedString0(i, 4);
    }

    /**
     * Returns a string representation of the integer argument as an
     * unsigned integer in base&nbsp;8.
     *
     * <p>The unsigned integer value is the argument plus 2<sup>32</sup>
     * if the argument is negative; otherwise, it is equal to the
     * argument.  This value is converted to a string of ASCII digits
     * in octal (base&nbsp;8) with no extra leading {@code 0}s.
     *
     * <p>The value of the argument can be recovered from the returned
     * string {@code s} by calling {@link
     * Integer#parseUnsignedInt(String, int)
     * Integer.parseUnsignedInt(s, 8)}.
     *
     * <p>If the unsigned magnitude is zero, it is represented by a
     * single zero character {@code '0'} ({@code '\u005Cu0030'});
     * otherwise, the first character of the representation of the
     * unsigned magnitude will not be the zero character. The
     * following characters are used as octal digits:
     *
     * <blockquote>
     * {@code 01234567}
     * </blockquote>
     *
     * These are the characters {@code '\u005Cu0030'} through
     * {@code '\u005Cu0037'}.
     *
     * @param   i   an integer to be converted to a string.
     * @return  the string representation of the unsigned integer value
     *          represented by the argument in octal (base&nbsp;8).
     * @see #parseUnsignedInt(String, int)
     * @see #toUnsignedString(int, int)
     * @since   JDK1.0.2
     */
    public static String toOctalString(int i) {
        return toUnsignedString0(i, 3);
    }

    /**
     * Returns a string representation of the integer argument as an
     * unsigned integer in base&nbsp;2.
     *
     * <p>The unsigned integer value is the argument plus 2<sup>32</sup>
     * if the argument is negative; otherwise it is equal to the
     * argument.  This value is converted to a string of ASCII digits
     * in binary (base&nbsp;2) with no extra leading {@code 0}s.
     *
     * <p>The value of the argument can be recovered from the returned
     * string {@code s} by calling {@link
     * Integer#parseUnsignedInt(String, int)
     * Integer.parseUnsignedInt(s, 2)}.
     *
     * <p>If the unsigned magnitude is zero, it is represented by a
     * single zero character {@code '0'} ({@code '\u005Cu0030'});
     * otherwise, the first character of the representation of the
     * unsigned magnitude will not be the zero character. The
     * characters {@code '0'} ({@code '\u005Cu0030'}) and {@code
     * '1'} ({@code '\u005Cu0031'}) are used as binary digits.
     *
     * @param   i   an integer to be converted to a string.
     * @return  the string representation of the unsigned integer value
     *          represented by the argument in binary (base&nbsp;2).
     * @see #parseUnsignedInt(String, int)
     * @see #toUnsignedString(int, int)
     * @since   JDK1.0.2
     */
    public static String toBinaryString(int i) {
        return toUnsignedString0(i, 1);
    }


    /**将integer转为一个无符号数字。
     * @param val  转换为数字的原始值int
     * @param shift  log2(转换的进制),2进制为1，8进制为3，16进制为4
     * @return
     */
    private static String toUnsignedString0(int val, int shift) {
        // assert shift > 0 && shift <=5 : "Illegal shift value";
    	// mag=32-val作为二进制时前面所有的0的个数=val二进制时需要的位数
        int mag = Integer.SIZE - Integer.numberOfLeadingZeros(val);
        //chars为标识转换成char数组需要的长度，最小是1（因为有可能val=0）
        int chars = Math.max(((mag + (shift - 1)) / shift), 1);
        char[] buf = new char[chars];  //创建chars大小的char数组buf
        //在buf数组中，从下标0开始，长度为chars，填入数值为val，进制为2的shift次方的无符号整数
        formatUnsignedInt(val, shift, buf, 0, chars);

        // 使用接管buf的特殊字符串构造器
        return new String(buf, true);
    }

    /**
     * 将一个long（视为无符号整数）放入字符缓存区
     * @param val the unsigned int to format
     * @param shift the log2 of the base to format in (4 for hex, 3 for octal, 1 for binary)
     * @param buf the character buffer to write to
     * @param offset the offset in the destination buffer to start at
     * @param len the number of characters to write
     * @return the lowest character  location used
     */
     static int formatUnsignedInt(int val, int shift, char[] buf, int offset, int len) {
        int charPos = len; //从最后一格开始写入
        int radix = 1 << shift;  //进制为2的shift次方
        int mask = radix - 1;  //a mod radix=a & radix-1  当radix是2的n次方时
        do {
        	//先--charPos,因为一开始为len
            buf[offset + --charPos] = Integer.digits[val & mask];  //对应的位为mod的结果
            val >>>= shift;  //相当于val=val/radix
        } while (val != 0 && charPos > 0);  //当val不为0 而且指针>0

        return charPos;
    }

    /**这两个char数组是给getChars方法调用的
     * 
     */
    final static char [] DigitTens = {
        '0', '0', '0', '0', '0', '0', '0', '0', '0', '0',
        '1', '1', '1', '1', '1', '1', '1', '1', '1', '1',
        '2', '2', '2', '2', '2', '2', '2', '2', '2', '2',
        '3', '3', '3', '3', '3', '3', '3', '3', '3', '3',
        '4', '4', '4', '4', '4', '4', '4', '4', '4', '4',
        '5', '5', '5', '5', '5', '5', '5', '5', '5', '5',
        '6', '6', '6', '6', '6', '6', '6', '6', '6', '6',
        '7', '7', '7', '7', '7', '7', '7', '7', '7', '7',
        '8', '8', '8', '8', '8', '8', '8', '8', '8', '8',
        '9', '9', '9', '9', '9', '9', '9', '9', '9', '9',
        } ;

    final static char [] DigitOnes = {
        '0', '1', '2', '3', '4', '5', '6', '7', '8', '9',
        '0', '1', '2', '3', '4', '5', '6', '7', '8', '9',
        '0', '1', '2', '3', '4', '5', '6', '7', '8', '9',
        '0', '1', '2', '3', '4', '5', '6', '7', '8', '9',
        '0', '1', '2', '3', '4', '5', '6', '7', '8', '9',
        '0', '1', '2', '3', '4', '5', '6', '7', '8', '9',
        '0', '1', '2', '3', '4', '5', '6', '7', '8', '9',
        '0', '1', '2', '3', '4', '5', '6', '7', '8', '9',
        '0', '1', '2', '3', '4', '5', '6', '7', '8', '9',
        '0', '1', '2', '3', '4', '5', '6', '7', '8', '9',
        } ;

        // I use the "invariant division by multiplication" trick to
        // accelerate Integer.toString.  In particular we want to
        // avoid division by 10.
        //
        // The "trick" has roughly the same performance characteristics
        // as the "classic" Integer.toString code on a non-JIT VM.
        // The trick avoids .rem and .div calls but has a longer code
        // path and is thus dominated by dispatch overhead.  In the
        // JIT case the dispatch overhead doesn't exist and the
        // "trick" is considerably faster than the classic code.
        //
        // TODO-FIXME: convert (x * 52429) into the equiv shift-add
        // sequence.
        //
        // RE:  Division by Invariant Integers using Multiplication
        //      T Gralund, P Montgomery
        //      ACM PLDI 1994
        //

    /**
     * Returns a {@code String} object representing the
     * specified integer. The argument is converted to signed decimal
     * representation and returned as a string, exactly as if the
     * argument and radix 10 were given as arguments to the {@link
     * #toString(int, int)} method.
     *
     * @param   i   an integer to be converted.
     * @return  a string representation of the argument in base&nbsp;10.
     */
    public static String toString(int i) {
        if (i == Integer.MIN_VALUE)
            return "-2147483648";
        int size = (i < 0) ? stringSize(-i) + 1 : stringSize(i);
        char[] buf = new char[size];
        getChars(i, size, buf);
        return new String(buf, true);
    }

    /**
     * Returns a string representation of the argument as an unsigned
     * decimal value.
     *
     * The argument is converted to unsigned decimal representation
     * and returned as a string exactly as if the argument and radix
     * 10 were given as arguments to the {@link #toUnsignedString(int,
     * int)} method.
     *
     * @param   i  an integer to be converted to an unsigned string.
     * @return  an unsigned string representation of the argument.
     * @see     #toUnsignedString(int, int)
     * @since 1.8
     */
    public static String toUnsignedString(int i) {
        return Long.toString(toUnsignedLong(i));
    }

    /**
     * Places characters representing the integer i into the
     * character array buf. The characters are placed into
     * the buffer backwards starting with the least significant
     * digit at the specified index (exclusive), and working
     * backwards from there.
     *
     * Will fail if i == Integer.MIN_VALUE
     */
    static void getChars(int i, int index, char[] buf) {
        int q, r;
        int charPos = index;
        char sign = 0;

        if (i < 0) {
            sign = '-';
            i = -i;
        }

        // Generate two digits per iteration
        while (i >= 65536) {
            q = i / 100;
        // really: r = i - (q * 100);
            r = i - ((q << 6) + (q << 5) + (q << 2));
            i = q;
            buf [--charPos] = DigitOnes[r];
            buf [--charPos] = DigitTens[r];
        }

        // Fall thru to fast mode for smaller numbers
        // assert(i <= 65536, i);
        for (;;) {
            q = (i * 52429) >>> (16+3);
            r = i - ((q << 3) + (q << 1));  // r = i-(q*10) ...
            buf [--charPos] = digits [r];
            i = q;
            if (i == 0) break;
        }
        if (sign != 0) {
            buf [--charPos] = sign;
        }
    }

    /**
     * 根据数组里面的值，进行大小判断，从而得到10进制下，有几位
     */
    final static int [] sizeTable = { 9, 99, 999, 9999, 99999, 999999, 9999999,
                                      99999999, 999999999, Integer.MAX_VALUE };

    // Requires positive x
    static int stringSize(int x) {
        for (int i=0; ; i++)
            if (x <= sizeTable[i])
                return i+1;
    }

    /**
     * <p>将string参数解析为一个带符号的整数，进制由第二个参数指定。
     * 字符串中的character必须是对应进制的数字（由Character.digit(char ch, int radix)返回的非负值决定），
     * 但是第一个字符可以是ASCII的负数记号{@code '-'} ({@code '\u005Cu002D'})来表明是负值
     * 或者 一个ASCII的正数记号{@code '+'}({@code '\u005Cu002B'})来表明是正值。
     * 返回结果的整数值。
     * 
     * <p>如果有下面之一的情况发生，抛出NumberFormatException
     * 
     * <ul>
     * <li>第一个参数是null或者长度为0
     *
     * <li>进制比Character.MIN_RADIX（2）小或者比Character.MAX_RADIX（36） 大
     *
     * <li>string的某一个字符不是特定进制的数字，除了string长度大于1而且第一个字符是负号或正号
     *
     * <li>代表字符串的值不是int类型的值
     * </ul>
     *
     * <p>例如：
     * <blockquote><pre>
     * parseInt("0", 10) returns 0
     * parseInt("473", 10) returns 473
     * parseInt("+42", 10) returns 42
     * parseInt("-0", 10) returns 0
     * parseInt("-FF", 16) returns -255
     * parseInt("1100110", 2) returns 102
     * parseInt("2147483647", 10) returns 2147483647
     * parseInt("-2147483648", 10) returns -2147483648
     * parseInt("2147483648", 10) throws a NumberFormatException  超出int的最大范围
     * parseInt("99", 8) throws a NumberFormatException  9不是8进制的数字
     * parseInt("Kona", 10) throws a NumberFormatException  英文不是10进制的数字
     * parseInt("Kona", 27) returns 411787  如果是27位，可以用英文
     * </pre></blockquote>
     * 
     * <p>可以发现，当radix大于10的时候，可以用a,b,c,d,z 代表10，11，12，13，35
     * 
     * <p>最高为36进制，0-9，a-z，其中一个z代表35
     *
     * @param      s   the {@code String} containing the integer
     *                  representation to be parsed
     * @param      radix   the radix to be used while parsing {@code s}.
     * @return     the integer represented by the string argument in the
     *             specified radix.
     * @exception  NumberFormatException 如果string不包含一个可解析的int
     */
    public static int parseInt(String s, int radix)
                throws NumberFormatException
    {
        /*
         * 警告：这个方法可能在虚拟机初始化时调用，早于IntegerCache的初始化，注意不要调用valueOf方法。
         */
    	
    	//排除s为null和radix不在范围内的情况。
        if (s == null) {
            throw new NumberFormatException("null");
        }

        if (radix < Character.MIN_RADIX) {
            throw new NumberFormatException("radix " + radix +
                                            " less than Character.MIN_RADIX");
        }

        if (radix > Character.MAX_RADIX) {
            throw new NumberFormatException("radix " + radix +
                                            " greater than Character.MAX_RADIX");
        }

        int result = 0;
        boolean negative = false;  //默认是正数
        int i = 0, len = s.length(); 
        //由于后面的result是对应的负值，正数的limit是-Integer.MAX_VALUE，负数的limit是Integer.MIN_VALUE
        int limit = -Integer.MAX_VALUE;
        int multmin;
        int digit;

        if (len > 0) {
            char firstChar = s.charAt(0);
            //注意ascii中的顺序：+  -  0  5  9  A  Z  a  z
            if (firstChar < '0') { // 这种情况，首字符不可能是数字和字母，只可能是正负号或者其他
                if (firstChar == '-') { //首字符为负号
                    negative = true; //这种情况为负数
                    limit = Integer.MIN_VALUE;
                } else if (firstChar != '+') //首字符为正号时，negative无需改变
                	//不是正负号时，抛出错误
                    throw NumberFormatException.forInputString(s);

                if (len == 1) // 到这里，首字符为正负号，但是这种情况，后面得有别的,不能len=1
                    throw NumberFormatException.forInputString(s);
                i++; //i为string中的坐标
            }
            //处理为正负号的首字符结束
            
            //multmin是保证 result *= radix 后result<limit
            //所以只要result<limit/radix后再乘以即可
            multmin = limit / radix;
            while (i < len) {
            	//result以负数增加，避免max_value附近的意外。
            	//digit为s的i处的char，在radix进制下的int值。如果对应的char超出radix的返回，digit=1
                digit = Character.digit(s.charAt(i++),radix);
                if (digit < 0) {
                	//digit为负数，说明超出radix范围，报错
                    throw NumberFormatException.forInputString(s);
                }
                if (result < multmin) {
                    throw NumberFormatException.forInputString(s);
                }
                result *= radix;
                if (result < limit + digit) {
                	//即这次的结果：result-digit<limit，说明超出limit了，报错
                    throw NumberFormatException.forInputString(s);
                }
                //每次迭代的结果是result=result*radix-digit
                result -= digit;
            }
        } else {
        	//如果长度为0，则抛出异常。
            throw NumberFormatException.forInputString(s);
        }
        //因为result是以负数形式出现的，所以是负数，直接返回，是正数，返回-result
        return negative ? result : -result;
    }

    /**
     * <p>将string参数解析为一个带符号的整数。
     * 字符串中的character必须是数字字符，
     * 但是第一个字符可以是ASCII的负数记号{@code '-'} ({@code '\u005Cu002D'})来表明是负值
     * 或者 一个ASCII的正数记号{@code '+'}({@code '\u005Cu002B'})来表明是正值。
     * 返回结果的整数值，就像使用parseInt(s,10)的结果一样。
     *
     * @param s    a {@code String} containing the {@code int}
     *             representation to be parsed
     * @return     the integer value represented by the argument in decimal.
     * @exception  NumberFormatException  if the string does not contain a
     *               parsable integer.
     */
    public static int parseInt(String s) throws NumberFormatException {
        return parseInt(s,10);
    }

    /**
     * <p>将string参数解析为一个无符号整数，进制由第二个参数指定。
     * 无符号整数通常将负数关联为大于max_value的整数。
     * 
     * <p>字符串中的字符必须都是特定进制的数字（由Character.digit(char, int)返回的非负值决定），
     * 除了首字符可以说ASCII的正号 {@code '+'} ({@code '\u005Cu002B'}。结果整数会被返回。
     * 
     * <p>如果以下情况之一出现，报错NumberFormatException
     * 
     * <ul>
     * <li>第一个参数是null或者长度为0
     *
     * <li>进制比Character.MIN_RADIX（2）小或者比Character.MAX_RADIX（36） 大
     *
     * <li>字符串的任何字符不是特定进制的数字，除非首字母是正号而且字符串长度大于1
     *
     * <li>代表字符串的值大于最大的无符号整数int，2<sup>32</sup>-1.
     *
     * </ul>
     * 
     * <p>这样就是说给的字符串代表的值为[0,2<sup>32</sup>-1]
     * 
     * <p>如果字符串代表[0,2<sup>31</sup>-1]，普通的int正数范围内，与普通情况一样。
     * 如果在[2<sup>31</sup>,2<sup>32</sup>-1]，会返回负数，负数的二进制表示结果和字符串转为二进制一样。
     *
     *
     * @param      s   the {@code String} containing the unsigned integer
     *                  representation to be parsed
     * @param      radix   the radix to be used while parsing {@code s}.
     * @return     the integer represented by the string argument in the
     *             specified radix.
     * @throws     NumberFormatException if the {@code String}
     *             does not contain a parsable {@code int}.
     * @since 1.8
     */
    public static int parseUnsignedInt(String s, int radix)
                throws NumberFormatException {
        if (s == null)  {
        	//排除s为null，报错
            throw new NumberFormatException("null");
        }

        int len = s.length();
        if (len > 0) {
            char firstChar = s.charAt(0);
            if (firstChar == '-') {
            	//排除首字符为负号，报错
                throw new
                    NumberFormatException(String.format("Illegal leading minus sign " +
                                                       "on unsigned string %s.", s));
            } else {
                if (len <= 5 || // Integer.MAX_VALUE 在进制为 Character.MAX_RADIX 有6位
                    (radix == 10 && len <= 9) ) { // Integer.MAX_VALUE 在10进制 由10位，2147483647 
                	//如果len<=5，保证对于所有进制都是正数。
                	//如果10进制，len <= 9，必定是正数
                	//正数情况下，可以视为普通的int情况，直接调用parseInt方法
                    return parseInt(s, radix);
                } else {
                	//这种情况下，s可能超过 Integer.MAX_VALUE，用long的方法解析
                    long ell = Long.parseLong(s, radix);  
                    if ((ell & 0xffff_ffff_0000_0000L) == 0) {
                    	//这个证明结果只有后32位存在1，前32位必定为0，在int的32位的范围内，可以强转为int
                        return (int) ell;
                    } else {
                        throw new
                            NumberFormatException(String.format("String value %s exceeds " +
                                                                "range of unsigned int.", s));
                    }
                }
            }
        } else {
        	//排除s长度为0，报错
            throw NumberFormatException.forInputString(s);
        }
    }

    /**
     * Parses the string argument as an unsigned decimal integer. The
     * characters in the string must all be decimal digits, except
     * that the first character may be an an ASCII plus sign {@code
     * '+'} ({@code '\u005Cu002B'}). The resulting integer value
     * is returned, exactly as if the argument and the radix 10 were
     * given as arguments to the {@link
     * #parseUnsignedInt(java.lang.String, int)} method.
     * 
     * <p>将string参数解析为一个无符号的整数。
     * 字符串中的character必须是数字字符，
     * 但是第一个字符可以是一个ASCII的正数记号{@code '+'}({@code '\u005Cu002B'})来表明是正值。
     * 返回结果的整数值，就像使用parseUnsignedInt(s,10)的结果一样。
     *
     * @param s   a {@code String} containing the unsigned {@code int}
     *            representation to be parsed
     * @return    the unsigned integer value represented by the argument in decimal.
     * @throws    NumberFormatException  if the string does not contain a
     *            parsable unsigned integer.
     * @since 1.8
     */
    public static int parseUnsignedInt(String s) throws NumberFormatException {
        return parseUnsignedInt(s, 10);
    }

    /**
     * Returns an {@code Integer} object holding the value
     * extracted from the specified {@code String} when parsed
     * with the radix given by the second argument. The first argument
     * is interpreted as representing a signed integer in the radix
     * specified by the second argument, exactly as if the arguments
     * were given to the {@link #parseInt(java.lang.String, int)}
     * method. The result is an {@code Integer} object that
     * represents the integer value specified by the string.
     *
     * <p>In other words, this method returns an {@code Integer}
     * object equal to the value of:
     *
     * <blockquote>
     *  {@code new Integer(Integer.parseInt(s, radix))}
     * </blockquote>
     *
     * @param      s   the string to be parsed.
     * @param      radix the radix to be used in interpreting {@code s}
     * @return     an {@code Integer} object holding the value
     *             represented by the string argument in the specified
     *             radix.
     * @exception NumberFormatException if the {@code String}
     *            does not contain a parsable {@code int}.
     */
    public static Integer valueOf(String s, int radix) throws NumberFormatException {
        return Integer.valueOf(parseInt(s,radix));
    }

    /**
     * Returns an {@code Integer} object holding the
     * value of the specified {@code String}. The argument is
     * interpreted as representing a signed decimal integer, exactly
     * as if the argument were given to the {@link
     * #parseInt(java.lang.String)} method. The result is an
     * {@code Integer} object that represents the integer value
     * specified by the string.
     *
     * <p>In other words, this method returns an {@code Integer}
     * object equal to the value of:
     *
     * <blockquote>
     *  {@code new Integer(Integer.parseInt(s))}
     * </blockquote>
     *
     * @param      s   the string to be parsed.
     * @return     an {@code Integer} object holding the value
     *             represented by the string argument.
     * @exception  NumberFormatException  if the string cannot be parsed
     *             as an integer.
     */
    public static Integer valueOf(String s) throws NumberFormatException {
        return Integer.valueOf(parseInt(s, 10));
    }

    /**
     * <p>缓存，支持由JLS要求的-128到127（包含）的值的自动装箱的对象语义标识。
     * 
     * <p>缓存在第一次使用时初始化。缓存的大小由选项 {@code -XX:AutoBoxCacheMax=<size>} 控制。
     * 在虚拟机初始化时，值IntegerCache.high可以被类sun.misc.VM中的私有系统属性设置并保存。
     * 
     * <p>注意；它是私有的static内部类，只能由Integer类访问
     */

    private static class IntegerCache {
        static final int low = -128;
        static final int high;
        static final Integer cache[];
        //第一次加载这个类的时候执行下面代码
        static {
            // high 值能够通过属性配置，但是默认是127
            int h = 127;
            //从sun.misc.VM得到high
            String integerCacheHighPropValue =
                sun.misc.VM.getSavedProperty("java.lang.Integer.IntegerCache.high");
            if (integerCacheHighPropValue != null) {
                try {
                	//如果设置了的话，127<=high<=Integer.MAX_VALUE - 129
                	//即如果不超过上下限，即为设置的值，否则为上下限
                    int i = parseInt(integerCacheHighPropValue);
                    i = Math.max(i, 127);
                    // 数组最大的长度是Integer.MAX_VALUE
                    h = Math.min(i, Integer.MAX_VALUE - (-low) -1);
                } catch( NumberFormatException nfe) {
                    // If the property cannot be parsed into an int, ignore it.
                }
            }
            high = h;
            //根据high和low创建integer数组cache
            cache = new Integer[(high - low) + 1];
            int j = low;
            for(int k = 0; k < cache.length; k++)
            	//cache中下标0对应-128，length-1对应high
                cache[k] = new Integer(j++);

            // 保证包含[-128, 127]  (JLS7 5.1.7)
            assert IntegerCache.high >= 127;
        }
        //私有的构造器，不能创建实例
        private IntegerCache() {}
    }

    /**
     * <p>返回一个代表特定int值的Integer实例。
     * 如果不需要一个新的Integer实例，这个方法应该比构造器 Integer(int)优先使用，
     * 因为通过缓存频繁地请求值，这个方法可能会显著地提高空间和时间的效率。
     * 
     * <p>这个方法通常情况下，会缓存包含-128到127的值，也可能缓存超过这个返回的值。
     * （如果设置了{@code -XX:AutoBoxCacheMax=<size>}，那就可能会超过）
     *
     * <p>注意：这个方法会在  Integer a=1 时被使用，用来包装int类型。变成 Integer a=Integer.valueOf(1)
     *
     * @param  i an {@code int} value.
     * @return an {@code Integer} instance representing {@code i}.
     * @since  1.5
     */
    public static Integer valueOf(int i) {
        if (i >= IntegerCache.low && i <= IntegerCache.high)
        	//如果i在[IntegerCache.low,IntegerCache.high]
        	//就返回IntegerCache.cache数组中缓存的Integer对象
            return IntegerCache.cache[i + (-IntegerCache.low)];
        //如果不在范围内，直接调用 Integer(int)构造器返回
        return new Integer(i);
    }

    /**
     * <p>Integer的值
     * 
     * <p>注意它是private，final的，
     * 别人不能对它修改，只能换一个Integer对象
     * 
     * <p>Integer最重要的值，代表了这个Integer对象。
     *
     * @serial
     */
    private final int value;

    /**
     * 构造一个新分配的Integer对象，它的值代表特定的int值。
     *
     * @param   value   the value to be represented by the
     *                  {@code Integer} object.
     */
    public Integer(int value) {
    	//直接设置value即可
        this.value = value;
    }

    /**
     * Constructs a newly allocated {@code Integer} object that
     * represents the {@code int} value indicated by the
     * {@code String} parameter. The string is converted to an
     * {@code int} value in exactly the manner used by the
     * {@code parseInt} method for radix 10.
     * 
     * 构造一个新分配的Integer对象，它的值代表特定有String参数表明的int值。
     * 这个字符串被转换为int值，用基数10，调用parseInt方法
     *
     * @param      s   the {@code String} to be converted to an
     *                 {@code Integer}.
     * @exception  NumberFormatException  if the {@code String} does not
     *               contain a parsable integer.
     * @see        java.lang.Integer#parseInt(java.lang.String, int)
     */
    public Integer(String s) throws NumberFormatException {
        this.value = parseInt(s, 10);
    }

    /**
     * Returns the value of this {@code Integer} as a {@code byte}
     * after a narrowing primitive conversion.
     * @jls 5.1.3 Narrowing Primitive Conversions
     */
    public byte byteValue() {
        return (byte)value;
    }

    /**
     * Returns the value of this {@code Integer} as a {@code short}
     * after a narrowing primitive conversion.
     * @jls 5.1.3 Narrowing Primitive Conversions
     */
    public short shortValue() {
        return (short)value;
    }

    /**
     * Returns the value of this {@code Integer} as an
     * {@code int}.
     */
    public int intValue() {
        return value;
    }

    /**
     * Returns the value of this {@code Integer} as a {@code long}
     * after a widening primitive conversion.
     * @jls 5.1.2 Widening Primitive Conversions
     * @see Integer#toUnsignedLong(int)
     */
    public long longValue() {
        return (long)value;
    }

    /**
     * Returns the value of this {@code Integer} as a {@code float}
     * after a widening primitive conversion.
     * @jls 5.1.2 Widening Primitive Conversions
     */
    public float floatValue() {
        return (float)value;
    }

    /**
     * Returns the value of this {@code Integer} as a {@code double}
     * after a widening primitive conversion.
     * @jls 5.1.2 Widening Primitive Conversions
     */
    public double doubleValue() {
        return (double)value;
    }

    /**
     * 返回一个代表这个Integer的String对象。
     * 这个Integer的值被转换成有符号的数字表示，并以字符串返回，就像调用toString(int)方法。
     *
     * @return  a string representation of the value of this object in
     *          base&nbsp;10.
     */
    public String toString() {
    	//调用了最开始的toString方法，以这个Integer的value作为参数
        return toString(value);
    }

    /**
     * 返回这个Integer的hashcode（其实返回的是这个integer的int值）
     *
     * @return  a hash code value for this object, equal to the
     *          primitive {@code int} value represented by this
     *          {@code Integer} object.
     *          
     */
    @Override
    public int hashCode() {
    	//调用下面的这个方法
        return Integer.hashCode(value);
    }

    /**
     * 返回对于一个int的hashcode，与Integer.hashCode()兼容。
     * 返回的hashcode就是参数value
     *
     * @param value the value to hash
     * @since 1.8
     *
     * @return a hash code value for a {@code int} value.
     */
    public static int hashCode(int value) {
        return value;
    }

    /**
     * 将这个对象与特定对象比较。
     * 当且仅当参数不为null，而且是一个Integer对象，而且有着与这个对象相同的int值，才返回true。
     *
     * @param   obj   the object to compare with.
     * @return  {@code true} if the objects are the same;
     *          {@code false} otherwise.
     */
    public boolean equals(Object obj) {
        if (obj instanceof Integer) {
        	//先判断参数是否为Integer类型
        	//参数强转成Integer，调用intValue得到int值，双方比较即可
            return value == ((Integer)obj).intValue();
        }
        return false;
    }

    /**
     * Determines the integer value of the system property with the
     * specified name.
     *
     * <p>The first argument is treated as the name of a system
     * property.  System properties are accessible through the {@link
     * java.lang.System#getProperty(java.lang.String)} method. The
     * string value of this property is then interpreted as an integer
     * value using the grammar supported by {@link Integer#decode decode} and
     * an {@code Integer} object representing this value is returned.
     *
     * <p>If there is no property with the specified name, if the
     * specified name is empty or {@code null}, or if the property
     * does not have the correct numeric format, then {@code null} is
     * returned.
     *
     * <p>In other words, this method returns an {@code Integer}
     * object equal to the value of:
     *
     * <blockquote>
     *  {@code getInteger(nm, null)}
     * </blockquote>
     *
     * @param   nm   property name.
     * @return  the {@code Integer} value of the property.
     * @throws  SecurityException for the same reasons as
     *          {@link System#getProperty(String) System.getProperty}
     * @see     java.lang.System#getProperty(java.lang.String)
     * @see     java.lang.System#getProperty(java.lang.String, java.lang.String)
     */
    public static Integer getInteger(String nm) {
        return getInteger(nm, null);
    }

    /**
     * Determines the integer value of the system property with the
     * specified name.
     *
     * <p>The first argument is treated as the name of a system
     * property.  System properties are accessible through the {@link
     * java.lang.System#getProperty(java.lang.String)} method. The
     * string value of this property is then interpreted as an integer
     * value using the grammar supported by {@link Integer#decode decode} and
     * an {@code Integer} object representing this value is returned.
     *
     * <p>The second argument is the default value. An {@code Integer} object
     * that represents the value of the second argument is returned if there
     * is no property of the specified name, if the property does not have
     * the correct numeric format, or if the specified name is empty or
     * {@code null}.
     *
     * <p>In other words, this method returns an {@code Integer} object
     * equal to the value of:
     *
     * <blockquote>
     *  {@code getInteger(nm, new Integer(val))}
     * </blockquote>
     *
     * but in practice it may be implemented in a manner such as:
     *
     * <blockquote><pre>
     * Integer result = getInteger(nm, null);
     * return (result == null) ? new Integer(val) : result;
     * </pre></blockquote>
     *
     * to avoid the unnecessary allocation of an {@code Integer}
     * object when the default value is not needed.
     *
     * @param   nm   property name.
     * @param   val   default value.
     * @return  the {@code Integer} value of the property.
     * @throws  SecurityException for the same reasons as
     *          {@link System#getProperty(String) System.getProperty}
     * @see     java.lang.System#getProperty(java.lang.String)
     * @see     java.lang.System#getProperty(java.lang.String, java.lang.String)
     */
    public static Integer getInteger(String nm, int val) {
        Integer result = getInteger(nm, null);
        return (result == null) ? Integer.valueOf(val) : result;
    }

    /**
     * Returns the integer value of the system property with the
     * specified name.  The first argument is treated as the name of a
     * system property.  System properties are accessible through the
     * {@link java.lang.System#getProperty(java.lang.String)} method.
     * The string value of this property is then interpreted as an
     * integer value, as per the {@link Integer#decode decode} method,
     * and an {@code Integer} object representing this value is
     * returned; in summary:
     *
     * <ul><li>If the property value begins with the two ASCII characters
     *         {@code 0x} or the ASCII character {@code #}, not
     *      followed by a minus sign, then the rest of it is parsed as a
     *      hexadecimal integer exactly as by the method
     *      {@link #valueOf(java.lang.String, int)} with radix 16.
     * <li>If the property value begins with the ASCII character
     *     {@code 0} followed by another character, it is parsed as an
     *     octal integer exactly as by the method
     *     {@link #valueOf(java.lang.String, int)} with radix 8.
     * <li>Otherwise, the property value is parsed as a decimal integer
     * exactly as by the method {@link #valueOf(java.lang.String, int)}
     * with radix 10.
     * </ul>
     *
     * <p>The second argument is the default value. The default value is
     * returned if there is no property of the specified name, if the
     * property does not have the correct numeric format, or if the
     * specified name is empty or {@code null}.
     *
     * @param   nm   property name.
     * @param   val   default value.
     * @return  the {@code Integer} value of the property.
     * @throws  SecurityException for the same reasons as
     *          {@link System#getProperty(String) System.getProperty}
     * @see     System#getProperty(java.lang.String)
     * @see     System#getProperty(java.lang.String, java.lang.String)
     */
    public static Integer getInteger(String nm, Integer val) {
        String v = null;
        try {
            v = System.getProperty(nm);
        } catch (IllegalArgumentException | NullPointerException e) {
        }
        if (v != null) {
            try {
                return Integer.decode(v);
            } catch (NumberFormatException e) {
            }
        }
        return val;
    }

    /**
     * Decodes a {@code String} into an {@code Integer}.
     * Accepts decimal, hexadecimal, and octal numbers given
     * by the following grammar:
     *
     * <blockquote>
     * <dl>
     * <dt><i>DecodableString:</i>
     * <dd><i>Sign<sub>opt</sub> DecimalNumeral</i>
     * <dd><i>Sign<sub>opt</sub></i> {@code 0x} <i>HexDigits</i>
     * <dd><i>Sign<sub>opt</sub></i> {@code 0X} <i>HexDigits</i>
     * <dd><i>Sign<sub>opt</sub></i> {@code #} <i>HexDigits</i>
     * <dd><i>Sign<sub>opt</sub></i> {@code 0} <i>OctalDigits</i>
     *
     * <dt><i>Sign:</i>
     * <dd>{@code -}
     * <dd>{@code +}
     * </dl>
     * </blockquote>
     *
     * <i>DecimalNumeral</i>, <i>HexDigits</i>, and <i>OctalDigits</i>
     * are as defined in section 3.10.1 of
     * <cite>The Java&trade; Language Specification</cite>,
     * except that underscores are not accepted between digits.
     *
     * <p>The sequence of characters following an optional
     * sign and/or radix specifier ("{@code 0x}", "{@code 0X}",
     * "{@code #}", or leading zero) is parsed as by the {@code
     * Integer.parseInt} method with the indicated radix (10, 16, or
     * 8).  This sequence of characters must represent a positive
     * value or a {@link NumberFormatException} will be thrown.  The
     * result is negated if first character of the specified {@code
     * String} is the minus sign.  No whitespace characters are
     * permitted in the {@code String}.
     *
     * @param     nm the {@code String} to decode.
     * @return    an {@code Integer} object holding the {@code int}
     *             value represented by {@code nm}
     * @exception NumberFormatException  if the {@code String} does not
     *            contain a parsable integer.
     * @see java.lang.Integer#parseInt(java.lang.String, int)
     */
    public static Integer decode(String nm) throws NumberFormatException {
        int radix = 10;
        int index = 0;
        boolean negative = false;
        Integer result;

        if (nm.length() == 0)
            throw new NumberFormatException("Zero length string");
        char firstChar = nm.charAt(0);
        // Handle sign, if present
        if (firstChar == '-') {
            negative = true;
            index++;
        } else if (firstChar == '+')
            index++;

        // Handle radix specifier, if present
        if (nm.startsWith("0x", index) || nm.startsWith("0X", index)) {
            index += 2;
            radix = 16;
        }
        else if (nm.startsWith("#", index)) {
            index ++;
            radix = 16;
        }
        else if (nm.startsWith("0", index) && nm.length() > 1 + index) {
            index ++;
            radix = 8;
        }

        if (nm.startsWith("-", index) || nm.startsWith("+", index))
            throw new NumberFormatException("Sign character in wrong position");

        try {
            result = Integer.valueOf(nm.substring(index), radix);
            result = negative ? Integer.valueOf(-result.intValue()) : result;
        } catch (NumberFormatException e) {
            // If number is Integer.MIN_VALUE, we'll end up here. The next line
            // handles this case, and causes any genuine format error to be
            // rethrown.
            String constant = negative ? ("-" + nm.substring(index))
                                       : nm.substring(index);
            result = Integer.valueOf(constant, radix);
        }
        return result;
    }

    /**
     * 数值上比较两个Integer，实际调用compare方法
     *
     * @param   anotherInteger   the {@code Integer} to be compared.
     * @return   如果这个Integer与anotherInteger相同，返回0；
     *           Integer小于anotherInteger，返回-1；
     *           Integer大于anotherInteger，返回1；
     * @since   1.2
     */
    public int compareTo(Integer anotherInteger) {
        return compare(this.value, anotherInteger.value);
    }

    /**
     * 数值上，比较两个int值。返回的值会与下面的语句返回的相同
     * <pre>
     *    Integer.valueOf(x).compareTo(Integer.valueOf(y))
     * </pre>
     *
     * @param  x the first {@code int} to compare
     * @param  y the second {@code int} to compare
     * @return   如果x==y，返回0。
     * 		   	  如果x<y,返回一个小于0的值，-1。
     * 			  如果x>y,返回一个大于0的值，1。
     * @since 1.7
     */
    public static int compare(int x, int y) {
        return (x < y) ? -1 : ((x == y) ? 0 : 1);
    }

    /**
     * 比较两个视为无符号整数的int值。
     *
     * @param  x the first {@code int} to compare
     * @param  y the second {@code int} to compare
     * @return the value {@code 0} if {@code x == y}; a value less
     *         than {@code 0} if {@code x < y} as unsigned values; and
     *         a value greater than {@code 0} if {@code x > y} as
     *         unsigned values
     * @since 1.8
     */
    public static int compareUnsigned(int x, int y) {
        return compare(x + MIN_VALUE, y + MIN_VALUE);
        //两者加上MIN_VALUE，相当于减去2147483648，再比较
        //这样无符号对应[0,2^32-1] 变成[-2^31.2^31-1]
    }

    /**
     * Converts the argument to a {@code long} by an unsigned
     * conversion.  In an unsigned conversion to a {@code long}, the
     * high-order 32 bits of the {@code long} are zero and the
     * low-order 32 bits are equal to the bits of the integer
     * argument.
     *
     * Consequently, zero and positive {@code int} values are mapped
     * to a numerically equal {@code long} value and negative {@code
     * int} values are mapped to a {@code long} value equal to the
     * input plus 2<sup>32</sup>.
     *
     * @param  x the value to convert to an unsigned {@code long}
     * @return the argument converted to {@code long} by an unsigned
     *         conversion
     * @since 1.8
     */
    public static long toUnsignedLong(int x) {
        return ((long) x) & 0xffffffffL;
    }

    /**
     * Returns the unsigned quotient of dividing the first argument by
     * the second where each argument and the result is interpreted as
     * an unsigned value.
     *
     * <p>Note that in two's complement arithmetic, the three other
     * basic arithmetic operations of add, subtract, and multiply are
     * bit-wise identical if the two operands are regarded as both
     * being signed or both being unsigned.  Therefore separate {@code
     * addUnsigned}, etc. methods are not provided.
     *
     * @param dividend the value to be divided
     * @param divisor the value doing the dividing
     * @return the unsigned quotient of the first argument divided by
     * the second argument
     * @see #remainderUnsigned
     * @since 1.8
     */
    public static int divideUnsigned(int dividend, int divisor) {
        // In lieu of tricky code, for now just use long arithmetic.
        return (int)(toUnsignedLong(dividend) / toUnsignedLong(divisor));
    }

    /**
     * Returns the unsigned remainder from dividing the first argument
     * by the second where each argument and the result is interpreted
     * as an unsigned value.
     *
     * @param dividend the value to be divided
     * @param divisor the value doing the dividing
     * @return the unsigned remainder of the first argument divided by
     * the second argument
     * @see #divideUnsigned
     * @since 1.8
     */
    public static int remainderUnsigned(int dividend, int divisor) {
        // In lieu of tricky code, for now just use long arithmetic.
        return (int)(toUnsignedLong(dividend) % toUnsignedLong(divisor));
    }


    // 位操作

    /**
     * 用二进制补码形式表示int值的位数
     *
     * @since 1.5
     */
    @Native public static final int SIZE = 32;

    /**
     * 用二进制补码形式表示整数值的字节数。
     * 一个字节为8个bit，一个int总共4个byte，32个bit
     *
     * @since 1.8
     */
    public static final int BYTES = SIZE / Byte.SIZE;

    /**
     * Returns an {@code int} value with at most a single one-bit, in the
     * position of the highest-order ("leftmost") one-bit in the specified
     * {@code int} value.  Returns zero if the specified value has no
     * one-bits in its two's complement binary representation, that is, if it
     * is equal to zero.
     *
     * @param i the value whose highest one bit is to be computed
     * @return an {@code int} value with a single one-bit, in the position
     *     of the highest-order one-bit in the specified value, or zero if
     *     the specified value is itself equal to zero.
     * @since 1.5
     */
    public static int highestOneBit(int i) {
        // HD, Figure 3-1
        i |= (i >>  1);
        i |= (i >>  2);
        i |= (i >>  4);
        i |= (i >>  8);
        i |= (i >> 16);
        return i - (i >>> 1);
    }

    /**
     * Returns an {@code int} value with at most a single one-bit, in the
     * position of the lowest-order ("rightmost") one-bit in the specified
     * {@code int} value.  Returns zero if the specified value has no
     * one-bits in its two's complement binary representation, that is, if it
     * is equal to zero.
     *
     * @param i the value whose lowest one bit is to be computed
     * @return an {@code int} value with a single one-bit, in the position
     *     of the lowest-order one-bit in the specified value, or zero if
     *     the specified value is itself equal to zero.
     * @since 1.5
     */
    public static int lowestOneBit(int i) {
        // HD, Section 2-1
        return i & -i;
    }

    /**
     * Returns the number of zero bits preceding the highest-order
     * ("leftmost") one-bit in the two's complement binary representation
     * of the specified {@code int} value.  Returns 32 if the
     * specified value has no one-bits in its two's complement representation,
     * in other words if it is equal to zero.
     *
     * <p>Note that this method is closely related to the logarithm base 2.
     * For all positive {@code int} values x:
     * <ul>
     * <li>floor(log<sub>2</sub>(x)) = {@code 31 - numberOfLeadingZeros(x)}
     * <li>ceil(log<sub>2</sub>(x)) = {@code 32 - numberOfLeadingZeros(x - 1)}
     * </ul>
     *
     * @param i the value whose number of leading zeros is to be computed
     * @return the number of zero bits preceding the highest-order
     *     ("leftmost") one-bit in the two's complement binary representation
     *     of the specified {@code int} value, or 32 if the value
     *     is equal to zero.
     * @since 1.5
     */
    public static int numberOfLeadingZeros(int i) {
        // HD, Figure 5-6
        if (i == 0)
            return 32;
        int n = 1;
        if (i >>> 16 == 0) { n += 16; i <<= 16; }
        if (i >>> 24 == 0) { n +=  8; i <<=  8; }
        if (i >>> 28 == 0) { n +=  4; i <<=  4; }
        if (i >>> 30 == 0) { n +=  2; i <<=  2; }
        n -= i >>> 31;
        return n;
    }

    /**
     * Returns the number of zero bits following the lowest-order ("rightmost")
     * one-bit in the two's complement binary representation of the specified
     * {@code int} value.  Returns 32 if the specified value has no
     * one-bits in its two's complement representation, in other words if it is
     * equal to zero.
     *
     * @param i the value whose number of trailing zeros is to be computed
     * @return the number of zero bits following the lowest-order ("rightmost")
     *     one-bit in the two's complement binary representation of the
     *     specified {@code int} value, or 32 if the value is equal
     *     to zero.
     * @since 1.5
     */
    public static int numberOfTrailingZeros(int i) {
        // HD, Figure 5-14
        int y;
        if (i == 0) return 32;
        int n = 31;
        y = i <<16; if (y != 0) { n = n -16; i = y; }
        y = i << 8; if (y != 0) { n = n - 8; i = y; }
        y = i << 4; if (y != 0) { n = n - 4; i = y; }
        y = i << 2; if (y != 0) { n = n - 2; i = y; }
        return n - ((i << 1) >>> 31);
    }

    /**
     * Returns the number of one-bits in the two's complement binary
     * representation of the specified {@code int} value.  This function is
     * sometimes referred to as the <i>population count</i>.
     *
     * @param i the value whose bits are to be counted
     * @return the number of one-bits in the two's complement binary
     *     representation of the specified {@code int} value.
     * @since 1.5
     */
    public static int bitCount(int i) {
        // HD, Figure 5-2
        i = i - ((i >>> 1) & 0x55555555);
        i = (i & 0x33333333) + ((i >>> 2) & 0x33333333);
        i = (i + (i >>> 4)) & 0x0f0f0f0f;
        i = i + (i >>> 8);
        i = i + (i >>> 16);
        return i & 0x3f;
    }

    /**
     * Returns the value obtained by rotating the two's complement binary
     * representation of the specified {@code int} value left by the
     * specified number of bits.  (Bits shifted out of the left hand, or
     * high-order, side reenter on the right, or low-order.)
     *
     * <p>Note that left rotation with a negative distance is equivalent to
     * right rotation: {@code rotateLeft(val, -distance) == rotateRight(val,
     * distance)}.  Note also that rotation by any multiple of 32 is a
     * no-op, so all but the last five bits of the rotation distance can be
     * ignored, even if the distance is negative: {@code rotateLeft(val,
     * distance) == rotateLeft(val, distance & 0x1F)}.
     *
     * @param i the value whose bits are to be rotated left
     * @param distance the number of bit positions to rotate left
     * @return the value obtained by rotating the two's complement binary
     *     representation of the specified {@code int} value left by the
     *     specified number of bits.
     * @since 1.5
     */
    public static int rotateLeft(int i, int distance) {
        return (i << distance) | (i >>> -distance);
    }

    /**
     * Returns the value obtained by rotating the two's complement binary
     * representation of the specified {@code int} value right by the
     * specified number of bits.  (Bits shifted out of the right hand, or
     * low-order, side reenter on the left, or high-order.)
     *
     * <p>Note that right rotation with a negative distance is equivalent to
     * left rotation: {@code rotateRight(val, -distance) == rotateLeft(val,
     * distance)}.  Note also that rotation by any multiple of 32 is a
     * no-op, so all but the last five bits of the rotation distance can be
     * ignored, even if the distance is negative: {@code rotateRight(val,
     * distance) == rotateRight(val, distance & 0x1F)}.
     *
     * @param i the value whose bits are to be rotated right
     * @param distance the number of bit positions to rotate right
     * @return the value obtained by rotating the two's complement binary
     *     representation of the specified {@code int} value right by the
     *     specified number of bits.
     * @since 1.5
     */
    public static int rotateRight(int i, int distance) {
        return (i >>> distance) | (i << -distance);
    }

    /**
     * Returns the value obtained by reversing the order of the bits in the
     * two's complement binary representation of the specified {@code int}
     * value.
     *
     * @param i the value to be reversed
     * @return the value obtained by reversing order of the bits in the
     *     specified {@code int} value.
     * @since 1.5
     */
    public static int reverse(int i) {
        // HD, Figure 7-1
        i = (i & 0x55555555) << 1 | (i >>> 1) & 0x55555555;
        i = (i & 0x33333333) << 2 | (i >>> 2) & 0x33333333;
        i = (i & 0x0f0f0f0f) << 4 | (i >>> 4) & 0x0f0f0f0f;
        i = (i << 24) | ((i & 0xff00) << 8) |
            ((i >>> 8) & 0xff00) | (i >>> 24);
        return i;
    }

    /**
     * Returns the signum function of the specified {@code int} value.  (The
     * return value is -1 if the specified value is negative; 0 if the
     * specified value is zero; and 1 if the specified value is positive.)
     *
     * @param i the value whose signum is to be computed
     * @return the signum function of the specified {@code int} value.
     * @since 1.5
     */
    public static int signum(int i) {
        // HD, Section 2-7
        return (i >> 31) | (-i >>> 31);
    }

    /**
     * Returns the value obtained by reversing the order of the bytes in the
     * two's complement representation of the specified {@code int} value.
     *
     * @param i the value whose bytes are to be reversed
     * @return the value obtained by reversing the bytes in the specified
     *     {@code int} value.
     * @since 1.5
     */
    public static int reverseBytes(int i) {
        return ((i >>> 24)           ) |
               ((i >>   8) &   0xFF00) |
               ((i <<   8) & 0xFF0000) |
               ((i << 24));
    }

    /**
     * Adds two integers together as per the + operator.
     *
     * @param a the first operand
     * @param b the second operand
     * @return the sum of {@code a} and {@code b}
     * @see java.util.function.BinaryOperator
     * @since 1.8
     */
    public static int sum(int a, int b) {
        return a + b;
    }

    /**
     * Returns the greater of two {@code int} values
     * as if by calling {@link Math#max(int, int) Math.max}.
     *
     * @param a the first operand
     * @param b the second operand
     * @return the greater of {@code a} and {@code b}
     * @see java.util.function.BinaryOperator
     * @since 1.8
     */
    public static int max(int a, int b) {
        return Math.max(a, b);
    }

    /**
     * Returns the smaller of two {@code int} values
     * as if by calling {@link Math#min(int, int) Math.min}.
     *
     * @param a the first operand
     * @param b the second operand
     * @return the smaller of {@code a} and {@code b}
     * @see java.util.function.BinaryOperator
     * @since 1.8
     */
    public static int min(int a, int b) {
        return Math.min(a, b);
    }

    /** use serialVersionUID from JDK 1.0.2 for interoperability */
    @Native private static final long serialVersionUID = 1360826667806852920L;
}
